The key to solving this puzzle is looking at the basic form of these
"twin" numbers, which is some number k = 1 + 10^n multiplied by some number
10^(n-1) <= a < 10^n. If ak is a perfect square, k must have some
repeated factor, since a<k. Searching the possible values of k for one
with a repeated factor eventually turns up the number 1 + 10^11 = 11^2
star 826446281. So, we set a=826446281 and ak = 9090909091^2 =
82644628100826446281, but this needs leading zeros to fit the pattern.
So, we multiply by a suitable small square (in this case 16) to get the
above answer.